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Higher Mathematics
Wave Fctn Past Papers Unit 3 Outcome 4
Written Questions
1. Express 8 cos x ◦ − 6 sin x ◦ in the form k cos(x ◦ + a◦ ) where k > 0 and 0 < a < 360.
[SQA]
Part
•1
•2
•3
•4
Marks
4
ss:
ic:
pd:
pd:
Level
C
Calc.
CR
Content
T13
Answer
10 cos(x ◦ + 36·9◦ )
U3 OC4
2001 P2 Q5
•1 k cos x cos a − k sin x sin a
explicitly
•2 k cos a = 8 and k sin a = 6
explicitly
3
• k = 10
•4 a = 36·9
expand k cos(x ◦ + a◦ )
compare coefficients
process
process
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3.
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7. Find the maximum value of cos x − sin x and the value of x for which it occurs in
the interval 0 ≤ x ≤ 2π .
[SQA]
Part
•1
•2
•3
•4
•5
•6
Marks
6
ss:
ic:
pd:
pd:
pd:
ic:
Level
A/B
Calc.
CN
Content
T14
Answer
max value
x = 7π
4
•1
•2
•3
•4
•5
•6
use e.g. k cos(x + a)
expand chosen rule
compare coefficients
process
process
interpret trig expression
√
2
when
U3 OC4
2000 P1 Q10
e.g. use k cos(x + a)
k cos x cos a − k sin x sin a
k cos√a = 1 and k sin a = 1
k= 2
tan a = 1, a = π4 (45◦ is bad form)
√
max. value = 2 when x = 7π
(do
4
not accept 45◦ )
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Higher Mathematics
8.
[SQA]
√
(a) Show that 2 cos(x ◦ + 30◦ ) − sin x ◦ can be written as 3 cos x ◦ − 2 sin x ◦ .
√
(b) Express 3 cos x ◦ − 2 sin x ◦ in the form k cos(x ◦ + α◦ ) where k > 0 and
0 ≤ α ≤ 360 and find the values of k and α.
(c) Hence, or otherwise, solve the equation 2 cos(x ◦ + 30◦ ) = sin x ◦ + 1,
0 ≤ x ≤ 360.
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9.
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Higher Mathematics
10. The displacement,
√ d units, of a wave after t seconds, is given by the formula
d = cos 20t◦ + 3 sin 20t◦ .
[SQA]
(a) Express d in the form k cos(20t ◦ − α◦ ), where k > 0 and 0 ≤ α ≤ 360.
(b) Sketch the graph of d for 0 ≤ t ≤ 18.
(c) Find, correct to one decimal place, the values of t, 0 ≤ t ≤ 18, for which the
displacement is 1·5 units.
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11. (a) Write sin(x) − cos(x) in the form k sin(x − a) stating the values of k and a
where k > 0 and 0 ≤ a ≤ 2π
[SQA]
(b) Sketch the graph of y = sin(x) − cos(x) for 0 ≤ x ≤ 2π , showing clearly the
graph’s maximum and minimum values and where it cuts the x -axis and the
y-axis.
Part
(a)
(b)
•1
•2
•3
•4
Marks
4
3
ss:
ic:
pd:
pd:
Level
C
C
Calc.
NC
NC
Content
T13
T15, T14
know to expand, and expand
compare coefficients
write down the value of k
process a
•5 ic: sketch a sine curve
•6 ic: int/com max. and min. values
•7 pd: process intercepts
Answer
√
2 sin(x − π4 )
sketch
U3 OC4
2002 P1 Q9
•1 k sin x cos a − k cos x sin a
explicitly
•2 k cos a = 1 and k sin a = 1
explicitly
√
•3 k = 2
•4 a = π4 accept in degrees
stated
stated
•5 correct shape of graph (i.e. sin) but
not passing through√the origin√
•6 graph lies between 2 and − 2
•7 ( π4 , 0), ( 5π
accept only
4 , 0), (0, −1)
answers in radians
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12.
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13.
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14.
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15.
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[END OF WRITTEN QUESTIONS]
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